Gram method for computing least squares approximate solution. Algorithm 12.1 in the textbook uses the QR factorization to compute the least squares approximate solution xˆ = A†b, where the m × n matrix A has linearly independent columns. It has a complexity of 2mn2 flops. In this exercise we consider an alternative method: First, form the Gram matrix G = AT A and the vector h = AT b; and then compute xˆ = G−1h (using algorithm 11.2 in the textbook). What is the complexity of this method? Compare it to algorithm 12.1. Remark. You might find that the Gram algorithm appears to be a bit faster than the QR method, but the factor is not large enough to have any practical significance. The idea is useful in situations where G is partially available and can be computed more efficiently than by multiplying A and its transpose.

Use the Euclidean algorithm to find the GCD of 3 + 9i and 7-i

6. Let A = {1, 2, 3, 4} and B = {5, 6, 7}. Let f = {(1, 5),(2, 5),(3, 6),(x, y)} where x ∈ A and y ∈ B are to be determined by you. (a) In how many ways can you pick x ∈ A and y ∈ B such that f is not a function? (b) In how many ways can you pick x ∈ A and y ∈ B such that f : A → B is onto? (c) In how many ways can you pick x ∈ A and y ∈ B such that f : A → B is not

Find the first four nonzero terms in a power series expansion about x = 0 for a general solution to the given differential equation

w'' - 6x² w' + w = 0

Find the first four nonzero terms in a power series expansion about x = 0 for a general solution to the given differential equation

(x² + 7)y'' + y = 0

A) A 50 gallon tank initially contains 10 gallons of fresh water. At t = 0 t = 0 a brine solution containing 1 pound of salt per gallon is poured into the tank at the rate of 4 gal/min., while the well-stirred mixture leaves the tank at the rate of 1 gal/min. Find the amount of salt in the tank at the moment of overflow.

B) A tank contains 100100 g of salt and 400400 L of water. Water that contains 1414 grams of salt per liter enters the tank at the rate 44 L/min. The solution is mixed and drains from the tank at the rate 66 L/min.

Let yy be the number of g of salt in the tank after tt minutes.

The differential equation for this situation would be:

dydt=dydt=     

Given the initial condition y(0)y(0) = 100 The particular solution would be

y(t)y(t)=    

K-TAB, a slow-release potassium chloride tablet, contains 750 mg of potassium chloride [KCl; MW = 74.5] in a wax/polymer matrix. How many milliequivalents of potassium chloride (round to the nearest whole number) are supplied by a 1 tablet dose given 3 times a day?

Manager T. C. Downs of Plum Engines, a producer of lawn mowers and leaf blowers, must develop an aggregate plan given the forecast for engine demand shown in the table. The department has a regular output capacity of 130 engines per month. Regular output has a cost of $60 per engine. The beginning inventory is zero engines. Overtime has a cost of $90 per engine.

Explain the geometric interpretation of exact differential equations. Talk about gradients, the multivariable chain rule, parametric curves and velocity or tangent vectors. What do these have to do with the condition for exactness of a differential equation? Use a specific example and draw pictures to elaborate.

Calculus w/ analytical geometry:

please be concise

- Fubini's Theorem: if a function is
continuous on the domain R, then the triple
integral can be evaluated in any order that
describes R.
(a) Explain the significance of this theorem.
(b) Provide an example to illustrate this
theorem.

- multi-integration
(a) Explain the purpose of changing variables
when double or triple integrating.
(b) Post an example illustrating such a change of variables.

Suppose you draw a card from a well- shuffled deck of 52 cards. Determine the following probabilities .

a. drawing a 9

b.drawing a 3 or king

c.drawing a spade

d.drawing a black card

e. DRAWING A RED KING

3) We have not forgotten Halmos Pal. In class I asked you what’s wrong with this “proof” of Halmos’ that all horses are the same color? It’s time to tell me what you found. (Try the web.)

consider the solid S bounded by the two cylinders x^2+y^2=3 and y^2+z^2=3 in R^3

a.Find the volume of S by setting up and evaluating a double integral.

b.Find the surface area of the solid S. You may use symmetry to simplify the computation.

NEW NUMBERS: A chemical plant stores spare parts for maintenance in a large warehouse. Throughout the working day, maintenance personnel go to the warehouse to pick up supplies needed for their jobs. The warehouse receives a request for supplies, on average, every three minutes. The average request requires 2.75 minutes to fill. Maintenance employees are paid $21.50 per hour and warehouse employees are paid $16 per hour. The warehouse operates 8 hours per day.

a) Based on the number of maintenance employees in the system, an 8 hour work day, and the given arrival and service rates. What is the system cost per day (to the nearest $) if there is only 1 warehouse employees working?

b) Based on the number of maintenance employees in the system, an 8 hour work day, and the given arrival and service rates. What is the system cost per day (to the nearest $) if there are 2 warehouse employees working?

c) Based on the number of maintenance employees in the system, an 8 hour work day, and the given arrival and service rates. What is the system cost per day (to the nearest $) if there are 3 warehouse employees working?

d) What is the optimal number of warehouse employees to staff the warehouse?